Existence And Multiplicity Of Solutions For Fractional Impulsive Differential Equations With Dirichlet Boundary Value Problems

In this thesis, the existence and multiplicity for two classes fractional impulsive differential equations boundary value problems are discussed by using the critical point theory.We construct the functional frameworks, and gain the sufficient conditions via the critical point theory, in order to get the existence and multiplicity of solutions for the corresponding problems.The thesis consists of four parts:In the first chapter, we introduce the research background and development status of impulsive differential equations、fractional impulsive differential equations and differential equations with P-Laplacian.Then our main works of this thesis are also stated briefly.In the second chapter,we present the basic knowledge of fractional calculus and critical point theory, then expound some basic definitions and theorems needed in this thesis.In the third chapter,we try to discuss the existence and multiplicity for a class fractional impulsive fractional differential equations boundary value problems via the critical point theory, under the conditions of weakening of Ambrosetti-Rabinowitz. Firstly, we get the sufficient conditions for a existence solution,which are proved by the existence theorem of critical points, when the nonlinearity F is subquadratic. Secondly, by using the Symmetric Mountain Pass theorem, we establish the multiplicity for the above boundary value problem, when the nonlinearity F is superquadratic.Finally, we study the multiplicity for the above boundary value problem, when the nonlinearity F is asymptotically quadratic. Some examples are illustrated for verifying the correctness of the conclusions.In the fourth chapter,we attempt to investigate the existence and multiplicity for the fractional impulsive differential equations boundary value problems with P-Laplacian by using the critical point theory on the basic of satisfying the Cerami condition. The firstly, We gain the sufficient conditions for a solution are applying by the saddle point theorem when the nonlinearity F is sublinear.The secondly,we get the existence of the infinitely many solutions when the nonlinearity F is superlinear by using the fountain theorem.The finally,we establish the sufficient conditions for the infinitely many solutions when the nonlinearity F is asymptotically linear. Some examples are illustrated for verifying the feasible of the conclusions.